The design for the interior of the dome is made up of six hexagonal
layers, each layer surmounting the one before. The creation of each
layer is based on a 360^{o} circle with the hexagon inscribed
within the circle. The six points for the base layer of the hexagon
are placed counter-clockwise along the circle at the following angles
0^{o}, 60^{o}, 120^{o}, 180^{o},
240^{o}, and 300^{o} relative to the positive
*x*-axis. To compute the point (*x*,*y*) along a
circle at a given angle *t*, use the formula *x*,*y*)
= (*r*cos*t*, *r*sin*t*)

The *z*-coordinate*z*-coordinate of 0 since the height at that
level is 0.00 cm. The radius is computed for each layer using its
height as *v* in the equation for the two
cones. So for example, the radius for level 1 is 5.38 cm since
its height is 1.70 cm. This is used to compute the first hexagon
layer. In order to compute one triangle layer one needs to know the
points along the circle for two hexagons: the one for the layer sought
and the one above. For example, when computing the base level, one
needs the angles mentioned above and the angles for the first triangle
layer which are 30^{o}, 90^{o}, 150^{o},
210^{o}, 270^{o}, and 330^{o}. The triangles
in this layer are formed by connecting the point at 0^{o} on
the base layer with the points at 30^{o} and 330^{o}
on the first layer, connecting the point at 60^{o} on the base
layer with the points at 30^{o} and 90^{o} on the
first layer, and so on. Both these sets of angles are standard in
computing the other five layers. This is because hexagons in
alternate layers have vertices at the same angles. Thus, the second
triangle layer uses the same angles as the base level and the third
layer uses the same angles as the first. This pattern continues for
all six layers.

Optical Illusion & Projection in Domes: A Study of Guarino Guarini's Santissima Sindone |